from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(736, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,16]))
pari: [g,chi] = znchar(Mod(49,736))
Basic properties
| Modulus: | \(736\) | |
| Conductor: | \(184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{184}(141,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 736.x
\(\chi_{736}(49,\cdot)\) \(\chi_{736}(81,\cdot)\) \(\chi_{736}(177,\cdot)\) \(\chi_{736}(209,\cdot)\) \(\chi_{736}(305,\cdot)\) \(\chi_{736}(561,\cdot)\) \(\chi_{736}(593,\cdot)\) \(\chi_{736}(625,\cdot)\) \(\chi_{736}(657,\cdot)\) \(\chi_{736}(721,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.22.14741666340843480753092741810452692992.1 |
Values on generators
\((415,645,97)\) → \((1,-1,e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 736 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)